Aplastic anemia is a rare autoimmune disorder in which the bone marrow production of blood cells is greatly decreased or absent. Symptoms include fatigue, weakness, tiny reddish-purple marks on the skin, abnormal bruising, and bleeding from the gums, nose, or intestine. While some cases of aplastic anemia are caused by medications, toxic exposures, or inherited genes, most often the cause remains unknown. The purpose of this study is to determine the safety and efficacy of combining two drugs, sirolimus and cyclosporine, for treating individuals with aplastic anemia that has not responded to other treatments.